Mathematics and Statistics · Ch 11 — Measures of Dispersion

Standard Deviation for a Frequency Distribution (Discrete and Continuous)

When data is given as a frequency distribution, each distinct value (or class mid-point) $x_i$ occurs $f_i$ times, so it must be weighted by its frequency.

Note

Standard Deviation — frequency distribution

$\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{N}}, \qquad \sigma^2 = \frac{\sum f_i x_i^{2}}{N} - \left(\frac{\sum f_i x_i}{N}\right)^{2}, \qquad N = \sum f_i$

For a discrete distribution, $x_i$ are the values themselves; for a continuous distribution, $x_i$ are the class mid-points (mid-point $= \tfrac12(\text{lower limit} + \text{upper limit})$ ).

Step-deviation method (for continuous data with equal class width). The mid-points are often large numbers, so we shift and scale them. Choose an assumed mean $A$ (a convenient mid-point) and the common class width $h$ , and set

d_i = \frac{x_i - A}{h}.

Then the mean and standard deviation are recovered by:

Note

Step-deviation formulas

$\bar{x} = A + h\left(\frac{\sum f_i d_i}{N}\right), \qquad \sigma = h\sqrt{\frac{\sum f_i d_i^{2}}{N} - \left(\frac{\sum f_i d_i}{N}\right)^{2}}$

The $h$ outside the root undoes the scaling; the shift by $A$ does not affect $\sigma$ (adding a constant never changes the spread), which is why $A$ appears in the mean but not in the standard deviation. …

Definition 12Standard deviation of a frequency distribution

$\sigma = \sqrt{\dfrac{\sum f_i(x_i-\bar{x})^2}{N}} = \sqrt{\dfrac{\sum f_i x_i^2}{N} - \bar{x}^{\,2}}$ , using values $x_i$ (discrete) or class mid-points $x_i$ (continuous …

Definition 13Step-deviation method

With $d_i = \dfrac{x_i - A}{h}$ ( $A$ = assumed mean, $h$ = class width): $\sigma = h\sqrt{\dfrac{\sum f_i d_i^2}{N} - \left(\dfrac{\sum f_i d_i}{N}\right)^2}$ ; a quick route to $\sigma$ for cont …

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